Given two starting numbers  , the following
table gives the unique sequences  that contain
no three-term arithmetic progressions.
| Sloane | sequence | | A003278 | 1, 2, 4, 5, 10, 11, 13, 14, 28,
29, 31, 32, ... | | A033156 | 1, 3, 4, 6, 10, 12, 13, 15, 28,
30, 31, 33, ... | | A033157 | 1, 4, 5, 8, 10, 13, 14, 17, 28,
31, 32, 35, ... | | A033158 | 1, 5, 6, 8, 12, 13, 17, 24, 27,
32, 34, 38, ... | | A033159 | 2, 3, 5, 6, 11, 12, 14, 15, 29,
30, 32, 33, ... | | A033160 | 2, 4, 5, 7, 11, 13, 14, 16, 29,
31, 32, 34, ... | | A033161 | 2, 5, 6, 9, 11, 14, 15, 18, 29,
32, 33, 36, ... | | A033162 | 3, 4, 6, 7, 12, 13, 15, 16, 30,
31, 33, 34, ... | | A033163 | 3, 5, 6, 8, 12, 14, 15, 17, 30,
32, 33, 35, ... | | A033164 | 4, 5, 7, 8, 13, 14, 16, 17, 31,
32, 34, 35, ... |
Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences."
Theor. Comput. Sci. 98, 163-197, 1992.
Erdős, P. and Turán, P. "On Some Sequences of Integers." J.
London Math. Soc. 11, 261-264, 1936.
Gerver, J.; Propp, J.; and Simpson, J. "Greedily Partitioning the Natural Numbers into Sets Free of Arithmetic Progressions." Proc. Amer. Math. Soc. 102,
765-772, 1988.
Guy, R. K. "Theorem of van der Waerden, Szemerédi's Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204-209,
1994.
Iacobescu, F. "Smarandache Partition Type and Other Sequences." Bull.
Pure Appl. Sci. 16E, 237-240, 1997.
Ibstedt, H. "A Few Smarandache Sequences." Smarandache Notions J. 8,
170-183, 1997.
Sloane, N. J. A. Sequences A003278/M0975, A033156, A033157, A033158, A033159, A033160, A033161, A033162, A033163, and A033164 in "The On-Line Encyclopedia of Integer Sequences."
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